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f(x, y) = x^{2} + xyz + z Find f_{x} at (1,1,1)
f_{x} = 2x + yz
Put (x,y,z) = (1,1,1)
f_{x} = 2 + 1 = 3.
Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons increased by:
The minimum point of the function f(x) = (x^{2/3}) – x is at
Correct Answer : a
Explanation : f(x) = (x^2/3)  x
f'(x) = 2/3(x^{1/2})  1
f"(x) = 1/3(x^{3/2})
For critical points. f′(x)=0
=> 2/3(x^{1/2})  1 = 0
f has minimum value of x = 1
If x=a(θ+ sin θ) and y=a(1cosθ), then dy/dx will be equal
The minimum value of function y = x^{2} in the interval [1, 5] is
y =x 2 is strictly increasing function on [1,5]
∴ y= x 2 has minimum value at x = 1 is 1.
The function f(x) = 2x^{3} – 3x^{2} – 36x + 2 has its maxima at
What should be the value of λ such that the function defined below is continuous at x = π/22?
By the given condition
Consider function f(x) =(x^{2}4)^{2} where x is a real number. Then the function has
If f where ai (i = 0 to n) are constants, then
 Euler’s theorem for homogeneous function
A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct exterma for the curve 3x^{4} – 16x^{3} – 24x^{2} + 37 is
∇ × ∇ × P, where P is a vector, is equal to
The value of the integral of the function g(x, y) = 4x^{3} + 10y^{4} along the straight line segment from the point (0, 0) to the point (1, 2) in the xy plane is
The equation of the line passing through (0,0) and (1,2) is y = 2x
Given y x, y ) = 4x^{3}+ 10y^{4} = 4x^{3} + 10(2x )^{4} = 4x^{3}+ 160xy
If is a differentiable vector function and f is a sufficient differentiable scalar function, then curl
The temperature field in a body varies according to the equation T(x,y) = x^{3}+4xy. The direction of fastest variation in temperature at the point (1,0) is given by
Among the following, the pair of vectors orthogonal to each other is
Then we say that they are orthogonal. Choice (c) is correct.
The directional derivative of the scalar function f(x, y, z) = x^{2} + 2y^{2} + z at the point P = (1,1, 2) in the direction of the vector
Required directional derivatives at P(1,1,1)
=2
The Gauss divergence theorem relates certain
If P, Q and R are three points having coordinates (3, –2, –1), (1, 3, 4), (2, 1, –2) in XYZ space, then the distance from point P to plane OQR (O being the origin of the coordinate system) is given by
The equation of the plane OQR is (O being origin).
Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product.
Then the determinant det
If a  b = 3 and a^{2} + b^{2} = 29, find the value of ab.
2ab = (a^{2} + b^{2})  (a  b)^{2}
= 29  9 = 20
ab = 10.
If a vector R(t) ^{→} has a constant magnitude, then
On analysing the given (a) option, we find that will give constant magnitude, so first
differentiation of the integration will be zero.
For the scalar field magnitude of the gradient at the point(1,3) is
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